3.4-3.5: Second-order mathematical models

What is an assumption of a spring mass system?

The top end of a vertically oriented spring is fixed

What is an assumption of a spring mass system?

An object of mass m is connected to the bottom end of the spring

What is an assumption of a spring mass system?

The mass can only move vertically

What is an assumption of a spring mass system?

The only forces acting on the mass are the resting force of the spring and the force of gravity

What is an assumption of a spring mass system?

The mass has initial displacement x0

What is an assumption of a spring mass system?

The mass has initial velocity v0

What is the type of solution in a spring mass system?

periodic

What is Hooke’s law?

Fs=-k(s+x)

What is Newton’s law?

Fn=ma

What is the equation of a spring mass system?

ma=-k(s+x)+mg

At rest in a spring mass system, what equals zero?

acceleration & displacement

What is the equation of a spring mass system at a=0?

mx’’+kx=0

At a=0, what is the equation of a spring mass system that has been divided by m?

ẍ+ω²x=0

What is ω² equal to in a spring mass system?

k/m

What is an assumption of a simple pendulum/swinging rod system?

A rod is allowed to swing freely with one fixed end.

What is an assumption of a simple pendulum/swinging rod system?

A bob with mass m is attached to the other end of the rod.

What is an assumption of a simple pendulum/swinging rod system?

The only forces acting on the bob are the force of gravity and the tension in the rod.

What is an assumption of a simple pendulum/swinging rod system?

The rod has length ℓ.

What is an assumption of a simple pendulum/swinging rod system?

The mass of the rod is negligible.

What is an assumption of a simple pendulum/swinging rod system?

The initial displacement angle of the rod is α.

What is an assumption of a simple pendulum/swinging rod system?

The initial rate of change in the rod’s displacement angle is β. ẋ

What is θ(t)?

The angle of the rod from the rest position at time t≥0.

What is θ?

The central angle of a circular sector.

What does θ=0 mean?

The rest position.

What is s(t)?

The angular position of the bob.

What is s(t) equal to?

ℓθ(t)

What does Newton’s law say about the force acting on the bob due to gravity?

F=-mg

What is an equation for the tangential force acting on the bob?

ms̈

What is an equation for the tangential force acting on the bob?

mℓθ``

What is an equation for the tangential force acting on the bob?

-(mg)sinθ

Under what condition can θ be used to approximate sinθ?

The displacement angle is small.

What is θ``?

The angular acceleration of the bob.

What is the equation of a simple pendulum problem?

θ``+(g/ℓ)sinθ=0

What is an equation of a linearized simple pendulum problem?

θ``+(g/ℓ)θ=0

What is an equation of a linearized simple pendulum problem?

θ``+(g/ℓ)θ=0

What is ω² equal to in a simple pendulum system?

g/ℓ

What is an equation of a linearized simple pendulum problem?

θ``+ω²θ=0

What is an assumption of a rotating string/jump rope system?

The endpoints of a sting are fixed at x=a and x=b.

What is an assumption of a rotating string/jump rope system?

The length of the string is slightly more than b-a.

What is an assumption of a rotating string/jump rope system?

The string is rotated about the x-axis (like a jump rope) at constant angular speed ω.

What is an assumption of a rotating string/jump rope system?

The string maintains a constant shape.

What is an assumption of a rotating string/jump rope system?

The magnitude of the tension in the string is constant along its length.

What is an assumption of a rotating string/jump rope system?

A constant horizontal force holds the ends of the string in place.

What is an assumption of a rotating string/jump rope system?

The string has linear mass density p (mass per unit length).

What is the name of the curve we are trying to find in a rotating string system?

Deflection curve

What is an equation of a rotating string system?

F=Tsinθ₂-Tsinθ₁

What is an equation of a rotating string system?

F=Ttanθ₂-Ttanθ₁

What is an equation of a rotating string system?

T[y’(x+Δx)-y’(x)]

What is an equation of a rotating string system DE?

y’’+(pω²/T)y(x)=0

How can y’’+(pω²/T)y=0 be rewritten?

y’’+μ²y=0

How is a Boundary Value Problem (BVP) different from an Initial Value Problem (IVP)?

They have boundary conditions instead of a initial condition; they can have a finite solution, no solution, or infinitely many solutions.

What type of differential equation problem is a rotating string system?

Boundary Value Problem

What is an assumption of a suspended wire system?

A wire is suspended between two points: A(a,b) and B(c,d).

What is an assumption of a suspended wire system?

The wire hangs under its own weight.

What is an assumption of a suspended wire system?

The tension in the wire at its lowest point is T₁.

What is an assumption of a suspended wire system?

The wire has linear mass density p (mass per unit length).

In a suspended wire system, what is the shape of the wire?

y(x)

In a suspended wire system, what is the lowest point?

P(u,v)

In a suspended wire system, what is a nearby point?

Q(x,y(x))

In a suspended wire system, what is the length of the wire from P to Q?

s(x)

In a suspended wire system, what is the equation of the horizontal forces?

T₁=T₂cosθ

In a suspended wire system, what is the equation of the vertical forces?

ps(x)g=T₂sinθ

What is the DE for a suspended wire system?

y’’(x)=(pg/T1) sqrt(1+[y’(x)]²)

What is an assumption for a electrical circuit?

A series circuit has a resistor, a capacitor, and an inductor connected in a single loop.

What is an assumption for a electrical circuit?

The impressed voltage (or emf) is E(t).

What is an assumption for a electrical circuit?

The resistance R, the capacitance C, and the inductance L are positive constants.

What is an equation for a electrical circuit?

Li’+Ri+(1/C)q=E(t)

What is an equation for a electrical circuit?

Lq’’+Rq’+(1/C)q=E(t)